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THE TRANSPORTATION OF LIQUIDS
119
To test the fundamental equation we will substitute therein the above computed
values: and find the speed:     ______________________________________
ua = 8.56 -^ 2 tan 18.5 + V32.2 X 168 V 0.685 + (8.56 -* 2 tan 18.5)2 = 102.6 ft. per second.
The calculated shut-off head would be H
102.62
= 164 ft. while the
20        64.4
actual shut-off head was 189 ft. showing that the above rule is only approximate. The hydraulic efficiency is the ratio of the useful power delivered in the water ( = WH) to the power delivered to the water by the impeller. The latter is equal to the torque multiplied by the angular velocity w. The torque is the mass W/g multiplied by the radius r and the tangential component ua  q of the absolute velocity wa, see Fig. 11. Hence,
Eh = WH +
- q)w = Hg + (ua - g)ua
Approximate rules: For short vanes, or open impellers: Eh = 30 per cent, for Volute Pumps Eh = 50 to 60 per cent, for Turbine Pumps Eh = 57 to 69 per cent.
TABLE 2.  APPROXIMATE RELATION BETWEEN vr/ua and aa
	Constant head			Variable head		
	aa	18	20	25	30	35
Volute pump    . .	?r_ Ua Vr Ua	0.08	0.125 0.085	0.15	0.175	0.2
						
Turbine pump ...				0.09	0.095	0.1
						
In example vr  8.56 ft. per second, and ua  102.6 ft. per second. This gives vr -r ua = 0.08, for which an angle of 18.5 was selected.
For a head of 200 ft. vr may be taken as high as 15 ft. per second. Peripheral Velocity and Rotative Speed
irdan    ___    dan .            = 229. 2ua
Utl ~ 12 X 60 ~ 229.2'         U ~~      da
Where n  revolutions per minute, da, external diameter, inches ua  peripheral velocity in feet per second.
The efficiency of a centrifugal pump falls off quite rapidly at speeds and capacities greater or less than those for which the pump is designed. Maximum efficiencies as high as 85 per cent have been attained under low heads (50 ft.), decreasing to about 75 per cent for high speed and high head (100 to 150 ft.) in single stage pumps and to from 55 to 70 per cent in two- and three-stage pumps for high lifts.
Overloading the Motor.-The usual characteristics show a rising power line, and it is important that the motor be selected that can take care of the maximum load. Thus in Fig. 11, although the normal conditions may call for only 149 hp. (point J), the maximum is 185 hp. (point L).
Tho advantages of centrifugal pumps over reciprocating pumps lie in their lower initial cost, smaller floor space requirements, less need for attendance, quiet operation, absence of excessive stresses in the pipe lines owing to the uniform discharge and adaptability for being driven by motors of high rotative speed. Silt or grit interferes but little with their working. The disadvantages are a certain lack of flexibility